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March 16, 2016, 01:49 
Solve 1D strong shock benchmark problems with 2ndorder FVM

#1 
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Yidong
Join Date: Nov 2011
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To solve the strong shock benchmark problems with 2ndorder finite volume method in 1D, for example,
(1) WoodwardColella blast waves <https://depts.washington.edu/clawpack/clawpack4.3/applications/euler/1d/wcblast/www/> (2) Sedov blast wave <http://wonka.physics.ncsu.edu/pub/VH1/bproblems.php> I know that with the class of "highresolution" schemes, we may get wellaccepted solution with only 500 cells. But if I have to stick to compact schemes, in which the slope reconstruction & limiting can only involve the faceneighboring cells of a cell, what could be the "best" choices so far? I started practicing with GreenGauss or leastsquares slope reconstruction plus minmax slope limiter. The result looks not so good even on a grid of 5000 cells. I appreciate any more comments that will be shared here. 

March 16, 2016, 04:58 

#2  
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Filippo Maria Denaro
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Quote:
have you checked the methods for the WC proposed in the book of LeVeque? 

May 21, 2016, 01:36 

#3  
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LUQILIN
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Quote:


May 21, 2016, 01:56 

#4  
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Yidong
Join Date: Nov 2011
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Quote:
Second, to solve Sedovblastwave problem using 2D triangular or 3D tetrahedral grids, I've found that the 2ndorder FVM (with linear gradient reconstruction + minmax slope limiter) has the difficulty to survive the initial steps. Later on I used a compactstencil WENO limiter, with which my code survived the initial extreme discontinuities. Thus I am wondering what approach have you tried to make a 2ndorder FVM robust enough to deal with a wide range of shock/blast/discontinuity problems in multidimensional unstructured grids. I am not interesting in structured meshes in 1D/2D/3D, as I know the limiting on the structured meshes are much better resolved. Thanks for any comment. 

May 22, 2016, 20:52 

#5  
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LUQILIN
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Quote:
Yang, Michael, and ZhiJian Wang. "A parameterfree generalized moment limiter for highorder methods on unstructured grids." Adv. Appl. Math. Mech 1.4 (2009): 451480. It can be applied to unstructured grids. If you are interested in high order family, maybe it will help you. I would like to know how you apply limiter strategy into boundary, say 1D case. I've tried WENO limiters and still been testing. The code blows. I upload my subroutine for trouble cell reconstruction. Its idea is from the paper: A simple weighted essentially nonoscillatory limiter for the CPR framework. Hope you could share your experiences and wisdom. 

May 22, 2016, 21:17 

#6 
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LUQILIN
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This is my subroutine of trouble cell's reconstruction using WENO limters.


May 22, 2016, 23:52 

#7  
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Yidong
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Quote:
o The compact WENO scheme I had used is based on the following paper. I believe there is no special treatment of the WENO limiter for a boundary cell in 1D/2D/3D. Though it has an oscillation indicator, yet it applies the very same algorithm to all the cells in the domain. Maybe I did not fully catch your described "blowup" issue, yet the only experience I could probably share about this is that you may want to play with some parameter in the WENO scheme to make it more diffusive. Luo, H., Baum, J. D., & Löhner, R. (2007). A Hermite WENObased limiter for discontinuous Galerkin method on unstructured grids. Journal of Computational Physics, 225(1), 686713. o I am also interested in the paper you introduced, and read part of the paper as for a quick overview. One concern I have is that the paper does not test the blast wave problems (which are of course much more challenging), e.g., the WoodwardCollela blast wave test, and the Sedov blast wave test. Maybe some later papers based on this limiter have done such verification. Did you happen to know the limitation of this limiter? Yang, Michael, and ZhiJian Wang. "A parameterfree generalized moment limiter for highorder methods on unstructured grids." Adv. Appl. Math. Mech 1.4 (2009): 451480. Thanks again for any comment. Best, 

May 23, 2016, 08:16 

#8 
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LUQILIN
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Thank you for your comments, Yidong.
1. I haven't test the minmod limiter（Yang, Michael, and ZhiJian Wang） for blast wave, yet could handle that as this type limiter is designed for SD, a high resolution method. Go through this paper, you may find what you want. Park J S, Chang T K, Kim C. Higherorder multidimensional limiting strategy for correction procedure via reconstruction[C]//52nd Aerospace Science Meeting. 2014: 20140772. 2. ＂I assume that [Tip No.1] is we should choose the primitivevariable based limiters over the conservativevariable limiters for those problems．＂ I think the following paper by CWang Shu could give you some proofs, A simple weighted essentially nonoscillatory limiter for the CPR framework. He found that schemes limiting Riemann Variables are more robust than those limiting conservative variables especially for highorder scheme.(order>=3) I used to the very WENO limiter in Shu's paper to solve a 1D nozzle shock problem. The exact solution of denstiy is in the attachment. I have no idea what kind of shock it is: strong, or weak? " Blows up" means when numerical solution reaches the extremum, the code diverges very quickly.But when numerical hasn't reach that far from the initial conditions, the code could successfully reconstruct the trouble cell. I don' t know where goes wrong and appreciate any comments. I would be very grateful. 3. I searched your name on google and happened to find that you interested key words are: DG, WENO, Compact Schemes, Unstructured Grids, etc. There's much similarity with my research field: CPR scheme. If you want to discuss with me please send me your email address, if you don't mind. Hope we can keep contact. Last edited by LUQILIN; May 23, 2016 at 09:28. 

May 23, 2016, 11:39 

#9  
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Yidong
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Quote:
o By the way, my main interests are 2nd/3rdorder DG and stabilization methods that are robust enough to be implemented in engineering codes. I do not purse the order higher than 3rd, as those higheroder methods are still far away from being appreciated and accepted in productionlevel code development in industry. o I know CPR a little bit, as I often came across ZJ Wang's and his students's presentations in previous AIAA meetings. It is my pleasure to extend the talk with you, in a separate email. Thanks, Yidong 

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